Abstract. We construct Markovian semigroups generated by noncommutative elliptic operators on a von Neumann algebra. We first introduce a general framework of quantum Feynman-Kac formula in terms of unitary evolutions and multipliers, and then apply the general result to our problem. We construct multipliers which are determined by operator valued stochastic differential equations and satisfy cocycle relations with respect to a randomized flow on a von Neumann algebra. Our works are greatly influenced by that of Lindsay and Sinha [LS]. In order to ensure Markovianity of the semigroups, we modify the method employed by Lindsay and Sinha [LS].